Conservation laws, solitary wave solutions, and lie analysis for the nonlinear chains of atoms

Nonlinear chains of atoms (NCA) are complex systems with rich dynamics, that influence various scientific disciplines. The lie symmetry approach is considered to analyze the NCA. The Lie symmetry method is a powerful mathematical tool for analyzing and solving differential equations with symmetries, facilitating the reduction of complexity and obtaining solutions. After getting the entire vector field by using the Lie scheme, we find the optimal system of symmetries. We have converted assumed PDE into nonlinear ODE by using the optimal system. The new auxiliary scheme is used to find the Travelling wave solutions, while graphical behaviour visually represents relationships and patterns in data or mathematical models. The multiplier method enables the identification of conservation laws, and fundamental principles in physics that assert certain quantities remain constant over time.

The analysis of nonlinear chains of atoms using Lie symmetry analysis and conservation laws is motivated by the desire to uncover the fundamental principles that govern the behaviour of these complex systems and gain a deeper understanding of their dynamics and properties. While there has been extensive research on nonlinear chains of atoms, there exist research gaps that can be addressed through the application of Lie symmetry analysis and conservation laws. These gaps include a need for comprehensive studies on the symmetries and conservation laws specific to nonlinear atomic chains, exploration of multiscale behaviour, and bridging the gap between theoretical analysis and practical applications. By addressing these research gaps, researchers can contribute to a more comprehensive understanding of nonlinear chains of atoms and pave the way for advancements in fields such as materials design, nanotechnology, and engineering applications.
The new auxiliary method 35,36 is a recently proposed method for solving challenging nonlinear PDEs. This method involves introducing an auxiliary variable and constructing a system of coupled equations involving both the original variables and the auxiliary variable. The resulting system of equations can be solved using numerical methods to obtain the solution to the original PDE. The new auxiliary method can handle highly nonlinear PDEs that are difficult to solve using other numerical methods, such as the finite difference approach or the finite element scheme. Additionally, this method can be used to obtain exact solutions to certain types of nonlinear PDEs, reducing the computational cost required to solve some types of nonlinear PDEs. Overall, the new auxiliary method is a promising tool for solving challenging nonlinear PDEs in various fields, including physics, engineering, and mathematical modelling.
Conservation laws of nonlinear PDEs [37][38][39][40] are essential concepts that relate to the principle of conservation of physical quantities like mass, energy, and momentum. These laws are expressed in terms of PDEs and have crucial importance in various fields, including engineering, physics, and mathematical modelling. They provide a mathematical framework to predict the behaviour of complex physical systems accurately and develop new theories and models to improve our understanding of the underlying physical mechanisms. Furthermore, conservation laws play a vital role in the design and analysis of physical systems and the development of numerical methods for solving challenging nonlinear PDEs, making them fundamental concepts in the study of nonlinear PDEs.
The layout of this research is presented as. Formation of supposed model is described in section 2. NAM and multiplier method are explained in section 3. technique and multiplier scheme are presented in section 2. Lie group analysis method is applied on supposed model and entire vector field is described in section 4. Optimal system, similarity reduction, wave solutions, and graphics are represented in section 5. Discussion of graphs and conservation laws of assumed model are described in section 6 and 7 respectively. The conclusions are stated in Sect. 8.

Formation of model
The Hamiltonian of the system is Foroutan et al. 41 : where m is the mass of the atom, V(|U n − U l |) stands for nonlinear potential and dot indicates for derivative w.r.t time. We consider l = 1 ± n and the subsequent potential: where h nl is relative displacement among l − th atom and n − th . The index i shows the distinct interactions via the particles. We omit our focus on the first and second neighbours. From Eqs. (1) and (2)  In Hamiltonian's equations, P .
n stands for generalized momentum. Assuming that the δ(inter-atom spacing) is small enough so that the continuum limit is reached, we substitute δ n → χ . Then with the subsequent constants; here in this paper, we will find out wave solutions and conservation laws for nonlinear Eq. (6) with the use of an appropriate transformation method.

Preliminaries
New auxiliary approach. Assuming the general form of partial PDE is of the form: where τ is the time part and χ is the spatial part and U = U(χ, τ ) is the dependent variable. We will follow the following steps.
Step 1 Suppose the new similarity variables or transformation is of the form where k and c both are actual parameters for Eq. (8). Putting the Eq. (9) into Eq. (8) and we get the new ODE below.
Step 2 Assume the general solution for Eq. (10) is of the form in the above solution, the C i 's are constants and we will fine later and the 1st ODE satisfied q(̺).
Step 3 In this step, we will use the balancing scheme to execute the value of N. For this, we have to compare the highest-order linear and nonlinear terms to find the value of N.
Step 4 Getting the coefficients of the powers of F q(̺) ( i = 0, 1, 2, 3, . . . ) by Eqs. (8), (11), and (12). Then collecting the terms of the same power and put them equal to zero which gives us a system of algebraic equations. After solving this system of equations by Maple.
Step 5 Finally we will get the different family of solutions for Eq. (12) of the form: Multiplier approach. Supposing the Eq. (8) and applying the following steps below: (1) Defining the total differential as: (2) Defining the Euler operator as below: Equation (39)  for some function U(µ 1 , µ 2 , . . . , µ m ).

Lie group analysis of Eq. (6)
Here, we are supposing the Lie approach for assumed Eq. (6). Now, suppose the one-parameter Lie group of infinitesimal transformations on (τ , χ , U) given by  (6), and X satisfy the Lie group conditions The fourth prolongation Pr (4) for X can be written as: furthermore, we have Let (x 1 , x 2 ) = (χ, τ ) , where D i can be written as: Substituting the values of η i which gives us the following vectors: We see that

Optimal system
In this section, we observe that from the obtained vector field Eq. (45), the X = {X 1 , X 2 } forms an abelian algebra. So we can use the (42) and get: Similarity reduction of Eq. (6). Here, we will find the similarity variables and analytical results for Eq. (6).
(46) Eq. (50) with the use of the proposed technique. Using the balancing method and we obtain N = 1 . Using the value of N = 1 in (11) and we have We have to put Eq. (51) into Eq. (50) and we get the system of the equation after comparing the coefficients of F q(̺) . With the use of Maple, we solved the obtained system of equations and got the following results.
Using Eq. (52) into Eq. (51) gives us the following set of solutions.
Where V 1 is an arbitrary constant. (52)

Graphics and discussion
A graphical representation of obtained solutions is discussed here in this section. By using the new auxiliary method we have constructed the analytical behaviour of the considered model in the form of trigonometric functions, hyperbolic trigonometric functions, and exponential, and algebraic-type results. The graph of the tangent function is periodic with a period of and has vertical asymptotes at odd multiples of 2 . As χ approaches these vertical asymptotes, the tangent function approaches positive or negative infinity depending on the direction of the approach. The graph of the cotangent function is also periodic with a period of and has horizontal asymptotes at even multiples of . As χ approaches these horizontal asymptotes, the cotangent function approaches zero. We have plotted the behaviour of some obtained results. Figure 1 shows the graphical behavior of u 1 (χ, τ ) for the choice of parameters Figure 2 represent the behaviour of u 2 (χ, τ ) for the choice of parameters Figure 3 shows The behaviour of U 19 shows Singularities exist in exponential functions due to their intrinsic nature of rapid and unbounded growth or decay. The exponential function, typically represented as f (χ) = e χ ,where e is Euler's number approximately equal to 2.71828,captures the exponential growth or decay phenomenon. However, certain values of χ result in problematic behaviour or undefined outcomes. When χ approaches positive or negative infinity, the exponential function exhibits an asymptotic behaviour, approaching infinity or zero, respectively, without reaching a definite value. These instances represent singularities where the exponential function becomes indeterminate or encounters difficulties in providing a well-defined result. These singularities in the exponential function highlight the inherent limitations and special characteristics associated with exponential growth or decay processes. (77)

Conservation laws
In this portion, we will construct the conservation laws by multiplier approach for Eq. (6). We obtain the determinant equation for �(X, t, u) by Eq. (41).
Using Eq. (38), we can write the Euler operator is of the form defining the total derivative operators D τ and D χ from Eq. (37).    by Eqs. (40) and (81), The following conservation laws are found Using Eq. (82) and we get the following two cases of conservation laws.

Conclusion
In this research, nonlinear chains of atoms(NCA) are studied. NCA play a crucial role in materials design and fabrication, enabling the development of advanced materials with tailored properties. The unique structural arrangements and interactions of nonlinear chains of atoms influence the mechanical, electrical, and optical properties of materials, which are essential for designing innovative technologies in fields such as nanotechnology, optoelectronics, and energy systems. Also, NCA are of great interest in the study of dynamical systems and nonlinear phenomena. They provide valuable insights into the behaviour of complex systems, allowing for the analysis and modelling of intricate dynamics observed in areas like physics, biology, and control theory. The Lie symmetry method and NAM enhances existing techniques, offering additional insights, improved accuracy, or simplified computations. Travelling wave solutions describe wave-like behaviour propagating through systems, while graphical behaviour provides visual representations of relationships and patterns in data or mathematical models. The multiplier method allows for the identification of conservation laws, which are fundamental principles in physics that state certain quantities remain constant over time. Understanding conservation laws and utilizing mathematical techniques such as the Lie symmetry method, travelling wave solutions, and graphical analysis contributes to a deeper understanding of nonlinear chains of atoms and their dynamics. (80)